Lorentzian Structure and Curvature Analysis of Osculating Type-2 Ruled Surfaces via the Type-2 Bishop Frame

Mohammed Messaoudi, Emad Solouma (), Mohammed N. Alshehri, Abdulrahman F. Aljohani and Marin Marin
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Mohammed Messaoudi: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Emad Solouma: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Mohammed N. Alshehri: Department of Mathematics, College of Arts Science, Najran University, Najran 66226, Saudi Arabia
Abdulrahman F. Aljohani: Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71411, Saudi Arabia
Marin Marin: Department of Mathematics and Computer Science, Transilvania, University of Brasov, 500036 Brasov, Romania

Mathematics, 2025, vol. 13, issue 21, 1-16

Abstract: This study investigates the geometry of osculating type-2 ruled surfaces in Minkowski 3-space E 1 3 , formulated through the Type-2 Bishop frame associated with a spacelike curve whose principal normal is timelike and binormal is spacelike. Using the hyperbolic transformation linking the Frenet–Serret and Bishop frames, we analyze how the Bishop curvatures ζ 1 and ζ 2 affect the geometric behavior and formation of such surfaces. Explicit criteria are derived for cylindrical, developable, and minimal configurations, together with analytical expressions for Gaussian and mean curvatures. We also determine the conditions under which the base curve behaves as a geodesic, asymptotic line, or line of curvature. Several illustrative examples in Minkowski 3-space are provided to visualize the geometric influence of ζ 1 and ζ 2 on flatness, minimality, and developability. Overall, the Type-2 Bishop frame offers a smooth and effective framework for characterizing Lorentzian geometry and symmetry of osculating ruled surfaces, extending classical Euclidean results to the Minkowski setting.

Keywords: Minkowski 3-space; Type-2 Bishop frame; osculating type-2 ruled surfaces; Lorentzian geometry; developable and minimal surfaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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